AI 中文总结
研究不可压缩连续体惯性运动，在几何和变分框架下扩展经典理论，证明相关流形性质，给出几何结构，推导出运动方程和度量，得到测地线流方程解的局部存在性结果，并通过例子对比可压缩与不可压缩情形。
AI 中文摘要
我们在几何和变分框架内研究不可压缩连续体的惯性运动，将经典的阿诺德 - 埃宾 - 马尔登理论从固定区域的保体积微分同胚群扩展到具有可变像的变形配置空间。在后一种情况下，由于缺乏群结构，需要证明一些在经典情形下直接可得的结果。我们表明具有适当正则性的保定向变形构成一个希尔伯特流形，保体积变形形成一个子流形，其切向量通过拉格朗日到欧拉图像对应映射到无散向量场。我们还给出了与可压缩连续体对应的几何结构。连续体的动能产生一个拉格朗日作用量，由此推导出惯性运动方程，以及一个度量，其测地线由惯性运动精确确定。虽然惯性运动仅对于不可压缩理想流体可与物理运动重合，但它可为一般连续体的配置流形提供自然参数化。我们得到了测地线流方程解的局部时间存在性的一般结果，并给出明确例子表明，对于相同初始数据，可压缩情形下连续体遵循的测地线可在有限时间内离开可允许变形流形，而相应的不可压缩测地线始终存在。
英文摘要
We study the inertial motion of incompressible continua within a geometric and variational framework, extending the classical Arnold-Ebin-Marsden theory from the group of volume-preserving diffeomorphisms of a fixed domain to configuration spaces of deformations with variable image. In the latter case, the lack of a group structure requires proving some results that are instead immediate in the classical setting. We show that the orientation-preserving deformations with suitable regularity constitute a Hilbert manifold and that volume-preserving deformations form a submanifold with tangent vectors that are mapped onto divergence-free vector fields by the Lagrangian-to-Eulerian-picture correspondence. In so doing, we also present the geometric structure corresponding to compressible continua. The kinetic energy of the continuum gives rise to both a Lagrangian action, from which the equations of inertial motion are deduced, and a metric, with geodesics that are identified precisely by inertial motions. While the inertial motion can coincide with a physical one only for incompressible perfect fluids, it can be used to provide a natural parametrization of the configuration manifold for generic continua. We obtain a general result of local-in-time existence of solutions for the geodesic flow equation and we present explicit examples showing that, for the same initial data, the geodesic followed by the continuum in the compressible case can leave the manifold of admissible deformations in finite time, while the corresponding incompressible geodesic exists for all times.
